Lower General Position Sets in Graphs

Abstract

A subset $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the \emph{lower general position number} $\gp ^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.

Figures (5)

• Figure 1: The Petersen graph with a maximum general position set (white) and a lower general position set (grey)
• Figure 2: The graph $G(7,6)$.
• Figure 3: The Petersen graph with a lower monophonic position set (grey)
• Figure 4: Construction for $a = 4$ and $b = 6$: the lower gp-set
• Figure 5: A graph $Z(2,2,1)$ with $\mathop{\mathrm{mp}}\nolimits^{-}(G) = 4$ (left) and $\mathop{\mathrm{gp}}\nolimits^-(G) = 3$ (right)

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Proposition 2.1
• proof
• Theorem 2.2
• Proposition 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2